CN110795800A - Screw rigidity determination method - Google Patents
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Abstract
The invention discloses a method for determining the rigidity of a screw, which at least comprises the following steps: decomposing the screw rod to obtain an equal-diameter rod taking the diameter of the root circle as the diameter and an equivalent spring with a thread structure; respectively calculating the axial stiffness, the torsional stiffness and the bending stiffness of the equal-diameter rod; respectively calculating the tensile stiffness, the torsional stiffness and the bending stiffness of the equivalent spring; superposing the axial stiffness of the equal-diameter rod and the tensile stiffness of the equivalent spring to obtain the axial total stiffness of the screw rod; superposing the torsional rigidity of the equal-diameter rod and the torsional rigidity of the equivalent spring to obtain the total torsional rigidity of the screw rod; and superposing the bending stiffness of the equal-diameter rod and the bending stiffness of the equivalent spring to obtain the total bending stiffness of the screw. According to the method for determining the rigidity of the screw, the influence of the thread in the equivalent spring on the rigidity of the screw can be fully considered when the rigidity of the screw is calculated, so that the calculation accuracy of the rigidity of the screw can be effectively improved.
Description
Technical Field
The invention relates to the technical field of screws, in particular to a screw rigidity determination method.
Background
The threaded connection structure is simple, the dismounting is convenient, and the connection mode is the most widely applied in mechanical structures. Forces in threaded connections are typically axial, torsional and bending moments. The corresponding stiffness is axial stiffness, torsional stiffness and bending stiffness.
At present, the existing method for calculating the rigidity of a screw is as follows: on the basis of theoretical analysis and calculation of screw connection, two methods of simulating screw connection by using spring elements and beam elements in assembly structure linear dynamics finite element analysis are provided, and then an assembly dynamics analysis model of a certain satellite payload part structural component is established under an MSC.
The following problems exist in calculating the screw stiffness using existing techniques: when the rigidity of the thread part is expressed, the small diameter of the thread is used as the diameter of the equivalent rod to express the rigidity, and the influence of the thread on the rigidity of the screw is not considered, so that the rigidity of the screw cannot be accurately obtained.
Disclosure of Invention
The embodiment of the invention aims to provide a method for determining the rigidity of a screw, which can fully consider the influence of threads on the rigidity of the screw when calculating the rigidity of the screw, so that the rigidity of the screw can be accurately calculated.
In order to solve the above problem, an embodiment of the present invention provides a method for determining a screw stiffness, which at least includes the following steps:
decomposing the screw rod to obtain an equal-diameter rod taking the diameter of the root circle as the diameter and an equivalent spring with a thread structure;
calculating the axial rigidity of the equal-diameter rod according to the axial force of the equal-diameter rod; calculating the torsional rigidity of the equal-diameter rod according to the relative torsional angle of the equal-diameter rod; calculating the bending rigidity of the equal-diameter rod according to the bending moment of the equal-diameter rod and the corner of the free end;
calculating the tensile stiffness of the equivalent spring according to the axial tension and compression load and the axial deformation of the equivalent spring; calculating the torsional rigidity of the equivalent spring according to the torque and the relative torsional angle of the equivalent spring; calculating the bending stiffness of the equivalent spring according to the bending deflection angle of the spring;
superposing the axial stiffness of the equal-diameter rod and the tensile stiffness of the equivalent spring to obtain the axial total stiffness of the screw rod; superposing the torsional rigidity of the equal-diameter rod and the torsional rigidity of the equivalent spring to obtain the total torsional rigidity of the screw rod; and superposing the bending stiffness of the equal-diameter rod and the bending stiffness of the equivalent spring to obtain the total bending stiffness of the screw rod.
Further, the obtaining of the axial stiffness of the equal-diameter rod according to the axial force of the equal-diameter rod includes:
calculating the elongation l of the equal-diameter rod according to the axial force as follows:
wherein F is the axial force, A is the cross-sectional area of the round rod, d is the diameter of the round section, E is the elastic modulus of the cantilever beam material, and L is the length of the round rod;
calculating the axial rigidity k of the equal-diameter rod according to the elongation1aComprises the following steps:
further, the calculating the torsional rigidity of the equal-diameter rod according to the relative torsional angle of the equal-diameter rod specifically includes:
setting the relative torsion angle of the equal-diameter rods
The expression is as follows:
wherein T is the torque applied, L is the length of the rod, and G is the shear of the materialShear modulus, d is the diameter of the circular cross-section, IpIs the polar moment of inertia of a round rod, an
Calculating the torsional rigidity k of the equal-diameter rod according to the torsional angle1bComprises the following steps:
further, the calculating the bending stiffness of the equal-diameter pole according to the bending moment of the equal-diameter pole and the corner of the free end includes:
according to the bending moment M borne by the cantilever beam with the circular section of the equal-diameter rod, calculating the rotation angle theta of the free end of the equal-diameter rod as follows:
wherein L is the length of the round rod, E is the elastic modulus of the cantilever beam material, d is the diameter of the round section, and I is the radial inertia moment of the cantilever beam,
calculating the bending rigidity k of the equal-diameter rod according to the rotating angle of the free end1cComprises the following steps:
further, the calculating of the tensile stiffness of the equivalent spring according to the axial tension and compression load and the axial deformation of the equivalent spring specifically includes:
according to the axial tension and compression load F of the equivalent spring, calculating the axial deformation F of the equivalent spring as follows:
calculating the tensile stiffness k of the equivalent spring according to the axial deformation2aComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe polar moment of inertia of the cross section of the spring material rotating around the p axis and the shear modulus of the G spring material.
Further, the calculating the torsional stiffness of the equivalent spring according to the torque and the relative torsion angle of the equivalent spring specifically includes:
calculating to obtain the relative torsion angle of the two ends of the equivalent spring according to the torque T received by the equivalent spring
Comprises the following steps:
calculating the torsional rigidity k according to the relative torsional angle2bComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe polar moment of inertia of the cross section of the spring material rotating around the p axis and the shear modulus of the G spring material.
Further, the calculating the bending stiffness of the equivalent spring according to the bending deflection angle of the spring specifically includes:
the bending deflection angle gamma of the equivalent spring is set as follows:
calculating the bending stiffness k of the spring from the bending deflection angle2cComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe spring material is the section polar moment of inertia of the spring material rotating around the p axis, the shear modulus of the G spring material and the height of the H spring.
The embodiment of the invention aims to provide a screw stiffness determining method, which can be used for decomposing a screw into a rod with equal diameter and an equivalent spring with a thread structure when calculating the stiffness of the screw, superposing the stiffness of the rod with equal diameter and the stiffness of the equivalent thread to obtain the overall stiffness of the screw, and fully considering the influence of the thread on the stiffness of the screw, thereby effectively improving the calculation accuracy of the stiffness of the screw.
Drawings
FIG. 1 is a schematic flow chart of a method for determining screw stiffness according to the present invention;
fig. 2 is a schematic structural diagram of an equivalent spring reference surface of the screw stiffness determination method provided by the invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 1, a method for determining a screw stiffness according to an embodiment of the present invention at least includes the following steps:
s1, decomposing the screw to obtain an equal-diameter rod taking the diameter of the root circle as the diameter and an equivalent spring with a thread structure;
s2, calculating the axial stiffness of the equal-diameter rod according to the axial force of the equal-diameter rod; calculating the torsional rigidity of the equal-diameter rod according to the relative torsional angle of the equal-diameter rod; calculating the bending rigidity of the equal-diameter rod according to the bending moment of the equal-diameter rod and the corner of the free end;
s3, calculating the tensile stiffness of the equivalent spring according to the axial tension and compression load and the axial deformation of the equivalent spring; calculating the torsional rigidity of the equivalent spring according to the torque and the relative torsional angle of the equivalent spring; calculating the bending stiffness of the equivalent spring according to the bending deflection angle of the spring;
s4, superposing the axial stiffness of the equal-diameter rod and the tensile stiffness of the equivalent spring to obtain the axial total stiffness of the screw; superposing the torsional rigidity of the equal-diameter rod and the torsional rigidity of the equivalent spring to obtain the total torsional rigidity of the screw rod; and superposing the bending stiffness of the equal-diameter rod and the bending stiffness of the equivalent spring to obtain the total bending stiffness of the screw.
In the embodiment of the invention, the screw is decomposed to obtain the equivalent springs with the equal-diameter rods and the thread structures, the rigidity of the equal-diameter rods and the rigidity of the equivalent springs are respectively calculated, the rigidity of the equal-diameter rods and the rigidity of the equivalent springs are superposed, the influence of the threads on the rigidity is fully considered, and the calculation precision of the rigidity of the screw can be effectively improved.
Preferably, the total axial rigidity of the screw is k in the embodiment of the inventionaThen, according to the stiffness superposition principle of the equivalent spring, the axial total stiffness k is obtainedaComprises the following steps: k is a radical ofa＝k1a+k2a(ii) a Setting the total torsional rigidity of the screw to kbThen, according to the stiffness superposition principle of the equivalent spring, the axial total stiffness k is obtainedbComprises the following steps: k is a radical ofb＝k1b+k2b(ii) a Let the bending stiffness of the screw be kcThen, according to the stiffness superposition principle of the equivalent spring, the axial total stiffness k is obtainedcComprises the following steps: k is a radical ofc＝k1c+k2c。
As a specific implementation manner of the embodiment of the present invention, the axial stiffness of the equal-diameter rod is obtained according to the axial force of the equal-diameter rod, specifically:
calculating the elongation l of the equal-diameter rod according to the axial force as follows:
wherein F is the axial force, A is the cross-sectional area of the round rod, d is the diameter of the round section, E is the elastic modulus of the cantilever beam material, and L is the length of the round rod;
calculating the axial rigidity k of the equal-diameter rod according to the elongation1aComprises the following steps:
in the embodiment of the invention, the axial rigidity of the equal-diameter rod can be accurately calculated through the axial force of the equal-diameter rod in the connecting structure.
As a specific implementation manner of the embodiment of the present invention, the torsional rigidity of the equal-diameter rod is calculated according to the relative torsional angle of the equal-diameter rod, specifically:
setting equal diameter rodRelative angle of torsion of
The expression is as follows:
wherein T is the applied torque, L is the length of the rod, G is the shear modulus of the material, d is the diameter of the circular cross-section, IpIs the polar moment of inertia of a round rod, an
Calculating the torsional rigidity k of the equal-diameter rod according to the torsional angle1bComprises the following steps:
in the embodiment of the invention, the torsion angle of the equal-diameter rod is obtained through the torsion force received by the equal-diameter rod in the connecting structure, and the torsion rigidity of the equal-diameter rod can be accurately calculated according to the torsion angle.
As a specific implementation manner of the embodiment of the present invention, the bending stiffness of the equal-diameter rod is calculated according to the bending moment of the equal-diameter rod and the corner of the free end, specifically:
according to the bending moment M received by the circular section cantilever beam of the equal-diameter rod, calculating the corner theta of the free end of the equal-diameter rod as follows:
wherein L is the length of the round rod, E is the elastic modulus of the cantilever beam material, d is the diameter of the round section, and I is the radial inertia moment of the cantilever beam,
according to selfCalculating the bending rigidity k of the equal-diameter rod according to the rotation angle of the end1cComprises the following steps:
in the embodiment of the invention, the corner theta of the free end of the equal-diameter rod is calculated through the bending moment of the equal-diameter rod in the connecting structure, and the bending rigidity of the equal-diameter rod can be accurately calculated according to the bending moment and the corner theta of the free end.
As a specific implementation manner of the embodiment of the present invention, the tensile stiffness of the equivalent spring is calculated according to the axial tension and compression load and the axial deformation of the equivalent spring, specifically:
according to the axial tension and compression load F of the equivalent spring, calculating the axial deformation F of the equivalent spring as follows:
calculating the tensile stiffness k of the equivalent spring according to the axial deformation2aComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe section polar moment of inertia of the spring material rotating around the p axis, and G is the shear modulus of the spring material.
Fig. 2 is a schematic view of a screw datum plane structure of a method for determining screw stiffness according to the present invention. In the embodiment of the invention, a tangent line of a central line of the spring material, which is taken at the center of the section A of the spring material, is a t axis, a normal line is an n axis, and a secondary normal line is a b axis. The axis T is positioned in a tangent plane T of the center line of the spring material, the axis n is positioned on the intersection line of the plane V and the plane V ', and the axis b is positioned in the plane V'. The equivalent spring corresponding to the thread structure is a cylindrical spring, and the section of the spring is the section of the thread.
As a specific implementation manner of the embodiment of the present invention, the torsional stiffness of the equivalent spring is calculated according to the torque and the relative torsional angle of the equivalent spring, specifically:
calculating the relative torsion angle of two ends of the equivalent spring according to the torque T received by the equivalent spring
Comprises the following steps:
calculating the torsional rigidity k according to the relative torsional angle2bComprises the following steps:
wherein l is the length of the helix, i.e., the deployed length of the effective working coil material of the spring, E is the modulus of elasticity of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe section polar moment of inertia of the spring material rotating around the p axis, and G is the shear modulus of the spring material.
As a specific implementation manner of the embodiment of the present invention, the bending stiffness of the equivalent spring is calculated according to the bending deflection angle of the spring, specifically:
the bending deflection angle γ of the equivalent spring is set as:
calculating the bending stiffness k of the spring from the bending deflection angle2cComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe cross-sectional polar moment of inertia of the spring material rotating around the p axis, G the shear modulus of the spring material, and H the spring height.
In the embodiment of the invention, the tensile stiffness, the torsional stiffness and the bending stiffness of the equivalent spring are calculated according to the elastic modulus, the inertia moment, the rotation angle and other data of the reference surface of the equivalent spring, and after the tensile stiffness, the torsional stiffness and the bending stiffness of the equivalent spring are accurately calculated, the tensile stiffness, the torsional stiffness and the bending stiffness of the equivalent spring are respectively superposed with the axial stiffness, the torsional stiffness and the bending stiffness of the equal-diameter rod, so that the overall stiffness of the screw is obtained, the influence of threads in the equivalent spring on the stiffness is fully considered, and the calculation precision of the stiffness of the screw can be effectively improved;
the embodiment of the invention has the following beneficial effects:
1. in the conventional manner, the influence of the screw thread on the overall rigidity is neglected. According to the embodiment of the invention, the screw rod is decomposed to obtain the equivalent springs with the equal-diameter rods and the thread structures, the rigidity of the equal-diameter rods and the rigidity of the equivalent springs are respectively calculated, the rigidity of the equal-diameter rods and the rigidity of the equivalent springs are superposed, the influence of the threads on the rigidity is fully considered, and the calculation precision of the rigidity of the screw rod can be effectively improved;
2. the embodiment of the invention is characterized in that according to the stress of the screw rod in the structure: axial force, torsional force and bending moment are respectively and comprehensively deduced from axial rigidity, torsional rigidity and bending rigidity, and the overall rigidity of the screw is discussed, so that a useful reference method can be effectively provided for subsequent engineering application and theoretical research;
3. the screw stiffness determination method provided by the embodiment of the invention has universality, the overall stiffness of the screw can be conveniently obtained only by determining the basic form and relevant parameters of the shape of the section of the screw thread, and important guiding significance can be provided for subsequent engineering application.
The foregoing is a preferred embodiment of the present invention, and it should be noted that it would be apparent to those skilled in the art that various modifications and enhancements can be made without departing from the principles of the invention, and such modifications and enhancements are also considered to be within the scope of the invention.
Claims (7)
1. A method of determining screw stiffness, comprising at least the steps of:
decomposing the screw rod to obtain an equal-diameter rod taking the diameter of the root circle as the diameter and an equivalent spring with a thread structure;
calculating the axial rigidity of the equal-diameter rod according to the axial force of the equal-diameter rod; calculating the torsional rigidity of the equal-diameter rod according to the relative torsional angle of the equal-diameter rod; calculating the bending rigidity of the equal-diameter rod according to the bending moment of the equal-diameter rod and the corner of the free end;
calculating the tensile stiffness of the equivalent spring according to the axial tension and compression load and the axial deformation of the equivalent spring; calculating the torsional rigidity of the equivalent spring according to the torque and the relative torsional angle of the equivalent spring; calculating the bending stiffness of the equivalent spring according to the bending deflection angle of the spring;
superposing the axial stiffness of the equal-diameter rod and the tensile stiffness of the equivalent spring to obtain the axial total stiffness of the screw rod; superposing the torsional rigidity of the equal-diameter rod and the torsional rigidity of the equivalent spring to obtain the total torsional rigidity of the screw rod; and superposing the bending stiffness of the equal-diameter rod and the bending stiffness of the equivalent spring to obtain the total bending stiffness of the screw rod.
2. The method for determining the rigidity of the screw according to claim 1, wherein the obtaining the axial rigidity of the constant-diameter rod according to the axial force of the constant-diameter rod comprises:
calculating the elongation l of the equal-diameter rod according to the axial force as follows:
wherein F is the axial force, A is the cross-sectional area of the round rod, d is the diameter of the round section, E is the elastic modulus of the cantilever beam material, and L is the length of the round rod;
calculating the axial rigidity k of the equal-diameter rod according to the elongation1aComprises the following steps:
3. the method for determining the stiffness of the screw according to claim 1, wherein the calculating the torsional stiffness of the equal-diameter rod according to the relative torsional angle of the equal-diameter rod comprises:
setting the relative torsion angle of the equal-diameter rods
The expression is as follows:
wherein T is the applied torque, L is the length of the rod, G is the shear modulus of the material, d is the diameter of the circular cross-section, IpIs the polar moment of inertia of a round rod, an
Calculating the torsional rigidity k of the equal-diameter rod according to the torsional angle1bComprises the following steps:
4. the method for determining the rigidity of the screw according to claim 1, wherein the calculating the bending rigidity of the constant-diameter bar according to the bending moment of the constant-diameter bar and the rotation angle of the free end comprises:
according to the bending moment M borne by the cantilever beam with the circular section of the equal-diameter rod, calculating the rotation angle theta of the free end of the equal-diameter rod as follows:
wherein L is the length of the round rod, E is the elastic modulus of the cantilever beam material, d is the diameter of the round section, and I is the radial inertia moment of the cantilever beam,
calculating the bending rigidity k of the equal-diameter rod according to the rotating angle of the free end1cComprises the following steps:
5. the screw stiffness determination method according to claim 1, wherein the calculating of the tensile stiffness of the equivalent spring from the axial tension and compression load and the axial deformation of the equivalent spring comprises:
according to the axial tension and compression load F of the equivalent spring, calculating the axial deformation F of the equivalent spring as follows:
calculating the tensile stiffness k of the equivalent spring according to the axial deformation2aComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe section polar moment of inertia of the spring material rotating around the p axis, and G is the shear modulus of the spring material.
6. The method for determining the stiffness of the screw according to claim 1, wherein the calculating the torsional stiffness of the equivalent spring according to the torque and the relative torsion angle of the equivalent spring comprises:
calculating to obtain the relative torsion angle of the two ends of the equivalent spring according to the torque T received by the equivalent springComprises the following steps:
calculating the torsional rigidity k according to the relative torsional angle2bComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe section polar moment of inertia of the spring material rotating around the p axis, and G is the shear modulus of the spring material.
7. The method for determining the stiffness of the screw according to claim 1, wherein the calculating the bending stiffness of the equivalent spring according to the bending deflection angle of the spring is specifically:
the bending deflection angle gamma of the equivalent spring is set as follows:
calculating the bending stiffness k of the spring from the bending deflection angle2cComprises the following steps:
wherein E is the elastic modulus of the spring material, InMoment of inertia, I, of spring material section revolving about axis nbIs the inertia moment of the spring material rotating around the b axis, α is the helical angle of the spring, D is the cylindrical diameter of the helical line, n is the number of helical coils, IpThe cross-sectional polar moment of inertia of the spring material rotating around the p axis, G the shear modulus of the spring material, and H the spring height.
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| CN111709144A (en) * | 2020-06-19 | 2020-09-25 | 中国核动力研究设计院 | Calculation method for calculating rigidity of compression spring |
| CN113434972A (en) * | 2021-06-15 | 2021-09-24 | 汕头大学 | Method for calculating axial static stiffness of planetary roller screw |
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| CN108362452A (en) * | 2018-02-04 | 2018-08-03 | 西北工业大学 | A kind of measurement method for the axial static dynamic stiffness measurement can be used for planetary roller screw |
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| JPH09218143A (en) * | 1996-02-13 | 1997-08-19 | Fujikura Rubber Ltd | Measuring method and device for shaft torsional rigidity distribution |
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| CN113434972B (en) * | 2021-06-15 | 2022-12-06 | 汕头大学 | Method for calculating axial static stiffness of planetary roller screw |
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